Question

Find the equation of the set of points which are equidistant from the points $(1,2,3)$ and $(3,2,-1)$

Answer 1

Let $P(x,y,z)$ be the point that is equidistant from points $A(1,2,3)$ and $B(3,2,-1)$ .

i.e. $PA=PB$

$\Rightarrow P{A}^2=P{B}^2$

$\Rightarrow {(x-1)}^2+{(y-2)}^2+{(z-3)}^2={(x-3)}^2+{(y-2)}^2+{(z+1)}^2$

$\Rightarrow x^2-2x+1+y^2-4y+4+z^2-6z+9=x^2-6x+9+y^2-4y+4+z^2+2z+1$

$\Rightarrow -2x-6z+14=-6x+2z+14$

$\Rightarrow -2x-6z+6x-2z=0$

$\Rightarrow 4x-8z=0$

$\Rightarrow x-2z=0$

Thus the required equation is $x-2z=0$.